A fun one with easy visualization is mechanical engineering statics, in which one calculates how much something deforms from a force (e.g., a bookshelf or bridge sags, a spring stretches, or an elastic tire or sponge or pillow squishes). The standard example is simple beam bending [1].
If you hold a thin beam (say, a ruler or a stiff piece of paper) horizontally in the air by one end, and press down on the free end, it bends from a straight line into a third-order (cubic) polynomial.
To calculate this, one considers the beam as a number of little segments connected to one another. The vertical force on each segment due to the force pressing down is constant over the beam. The first integral of this is the torque ("moment") on each segment. The second integral is the slope of the beam, and the third integral is the actual shape of the bent beam. If you consider it the other way around, the force is the third derivative of the resulting beam shape. This is often visualized in a "shear force and bending moment diagram".
The best part is there's a stupidly simple approximation to calculate how much bending you get from a single force (Hooke's law [2]): the distance the beam moves is proportional to the force (by some constant you can get either with these derivative calculations or by experiment).
Someone else pointed out Bézier curves. What is interesting is such are same as the bending equation!
Back in the pre-cad days, designers would bend a piece of thin, elastic spline-wood to draw curves. From the side it looks exactly like a Bézier curve, this is not a coincidence, Beziers are more or less same as the physical bending equation for such a spline.
If you hold a thin beam (say, a ruler or a stiff piece of paper) horizontally in the air by one end, and press down on the free end, it bends from a straight line into a third-order (cubic) polynomial.
To calculate this, one considers the beam as a number of little segments connected to one another. The vertical force on each segment due to the force pressing down is constant over the beam. The first integral of this is the torque ("moment") on each segment. The second integral is the slope of the beam, and the third integral is the actual shape of the bent beam. If you consider it the other way around, the force is the third derivative of the resulting beam shape. This is often visualized in a "shear force and bending moment diagram".
The best part is there's a stupidly simple approximation to calculate how much bending you get from a single force (Hooke's law [2]): the distance the beam moves is proportional to the force (by some constant you can get either with these derivative calculations or by experiment).
[1] https://en.wikipedia.org/wiki/Bending [2] https://en.wikipedia.org/wiki/Hooke%27s_law